## Banach Journal of Mathematical Analysis

### Lower and upper local uniform $K$-monotonicity in symmetric spaces

Maciej Ciesielski

#### Abstract

Using the local approach to the global structure of a symmetric space $E$, we establish a relationship between strict $K$-monotonicity, lower (resp., upper) local uniform $K$-monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform $K$-monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 314-330.

Dates
Accepted: 12 April 2017
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.bjma/1513674116

Digital Object Identifier
doi:10.1215/17358787-2017-0047

Mathematical Reviews number (MathSciNet)
MR3779716

Zentralblatt MATH identifier
06873503

#### Citation

Ciesielski, Maciej. Lower and upper local uniform $K$ -monotonicity in symmetric spaces. Banach J. Math. Anal. 12 (2018), no. 2, 314--330. doi:10.1215/17358787-2017-0047. https://projecteuclid.org/euclid.bjma/1513674116

#### References

• [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988.
• [2] Y. A. Brudnyi and N. Y. Kruglyak Interpolation Functors and Interpolation Spaces, I, North-Holland Math. Library 47, North-Holland, Amsterdam, 1991.
• [3] A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.
• [4] J. Cerdà, H. Hudzik, A. Kamińska, and M. Mastyło, Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces, Positivity 2 (1998), no. 4, 311–337.
• [5] V. I. Chilin, P. G. Dodds, A. A. Sedaev, and F. A. Sukochev, Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions, Trans. Amer. Math. Soc. 348, no. 12 (1996), 4895–4918.
• [6] V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31 (1994), no. 1, 35–65.
• [7] M. Ciesielski, On geometric structure of symmetric spaces, J. Math. Anal. Appl. 430 (2015), no. 1, 98–125.
• [8] M. Ciesielski, Hardy-Littlewood-Pólya relation in the best dominated approximation in symmetric spaces, J. Approx. Theory 213 (2017), 78–91.
• [9] M. Ciesielski, Strict $K$-monotonicity and $K$-order continuity in symmetric spaces, Positivity, published electronically 28 October 2017.
• [10] M. Ciesielski, P. Kolwicz, and A. Panfil, Local monotonicity structure of symmetric spaces with applications, J. Math. Anal. Appl. 409 (2014), no. 2, 649–662.
• [11] M. Ciesielski, P. Kolwicz, and R. Płuciennik, A note on strict $K$-monotonicity of some symmetric function spaces, Comment. Math. 53 (2013), no. 2, 311–322.
• [12] M. Ciesielski, P. Kolwicz, and R. Płuciennik, Local approach to Kadec-Klee properties in symmetric function spaces, J. Math. Anal. Appl. 426 (2015), no. 2, 700–726.
• [13] M. M. Czerwińska and A. Kamińska, Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators, Studia Math. 201 (2010), no. 3, 253–285.
• [14] H. Hudzik, A. Kamińska, and M. Mastyło, On geometric properties of Orlicz-Lorentz spaces, Canad. Math. Bull. 40 (1997), no. 3, 316–329.
• [15] A. Kamińska and L. Maligranda, On Lorentz spaces $\Gamma_{p,w}$, Israel J. Math. 140 (2004), 285–318.
• [16] A. Kamińska and L. Maligranda, Order convexity and concavity of Lorentz spaces $\Lambda_{p,w}$, $0<p<\infty$, Studia Math. 160 (2004), no. 3, 267–286.
• [17] S. G. Krein, Y. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, 1982.
• [18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II: Function Spaces, Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979.
• [19] G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), 411–429.
• [20] G. Pólya and G. Szegő, Problems and Theorems in Analysis, I: Series, Integral Calculus, Theory of Functions, Grundlehren Math. Wiss. 193, Springer, New York, 1972.
• [21] A. Sparr, “On the conjugate space of the Lorentz space $L(\phi,q)$” in Interpolation Theory and Applications, Contemp. Math. 445, Amer. Math. Soc., Providence, 2007, 313–336.